Given a compound waveform, it is desirable to accurately measure the waveform and its components, which may have been spawned by several sources. This is difficult when the waveform includes signals produced by different sources overlapping in time and frequency, low energy signals eclipsed by higher energy signals, rapid changes in frequency, and/or rapid changes in amplitude. If these waveforms could be more accurately measured and analyzed, it would greatly increase our ability to understand what they contain and how to modify them.
Analysis of waveforms is traditionally accomplished in the time and frequency domains. Typically, these waveforms are first captured digitally as amplitude samples in time, then a series of transforms is used to measure the signals and the result is displayed in a matrix. A variety of techniques have been developed to extract frequency/amplitude information from the time-series data. However, representing how the frequency and amplitude change with respect to time can be challenging, particularly when there are abrupt frequency and/or amplitude changes, or signals from multiple sources occupy the same time and frequency regions.
One common transform for obtaining time, frequency, and amplitude information is the Discrete Fourier Transform (DFT). Unfortunately, there is a tradeoff between frequency and time resolutions resulting from the size (dimension) of the DFT. The time window inspected by a DFT is proportional to its dimension. Thus, a large dimension DFT inspects a larger time window than a small dimension DFT. This larger time window makes a large dimension DFT slow to react to dynamic changes.
Conversely, a large dimension DFT slices up the frequency range into finer pieces. The maximum frequency measured by the DFT is half the sampling rate of the digitized signal. A DFT of dimension X divides frequency range from 0 to the max into X/2 equal sized “bins.” Thus, the size of each frequency bin in a DFT is equal to two times the sampling rate divided by its dimension.
So, higher dimension DFTs have higher frequency resolution but lower time resolution. Lower dimension DFTs have higher time resolution but lower frequency resolution. Because of this tradeoff, practitioners have sought modified DFTs or other alternative methods to accurately represent dynamic, time-varying waveforms with good resolution in both time and frequency.
Spectrograms, discrete wavelet transforms, Wigner-Ville, Gabor, and windowing techniques are just some of the techniques used to try to create more accurate time-frequency representations of dynamic signals that vary in time, frequency, and amplitude. Each technique has its own strengths and weaknesses: computational complexity (i.e., processing burden), artifacts/distortion, time accuracy and resolution, frequency accuracy and resolution, and amplitude accuracy and resolution. This invention is not a new transform technique, but a way of combining multiple transforms using any existing (or new) techniques.
This invention should not be confused with a vaguely similar but less sophisticated form of combining multiple FT data called spectral correlation. With spectral correlation, as found in the prior art, the same size FT is calculated, and the FT data are not taken simultaneously. Therefore, in contrast to this invention, spectral correlation does not take advantage of the simultaneous improvement in time and frequency resolution and accuracy. It also smears information over time, something this technique is designed to avoid.
The inventors have been issued several patents, which are hereby incorporated by reference. They are: Fast Find Fundamental Method, U.S. Pat. No. 6,766,288 B1; Method of Modifying Harmonic Content of a Complex Waveform, U.S. Pat. No. 7,003,120 B1; and Method of Signal Shredding, U.S. Pat. No. 6,798,886 B1. Provisional Application No. 61/118,198 filed Nov. 26, 2008 is also hereby incorporated by reference.